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<div  id='write'  class = 'is-node show-fences-line-number'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#header-n2">1. 引言(Introduction)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#header-n3">1.1 Welcome</a></span><span class="md-toc-item md-toc-h2" data-ref="n57"><a class="md-toc-inner" href="#header-n57">1.2 什么是机器学习(What is Machine Learning)</a></span><span class="md-toc-item md-toc-h2" data-ref="n113"><a class="md-toc-inner" href="#header-n113">1.3 监督学习(Supervised Learning)</a></span><span class="md-toc-item md-toc-h2" data-ref="n142"><a class="md-toc-inner" href="#header-n142">1.4 无监督学习(Unsupervised Learning)</a></span><span class="md-toc-item md-toc-h1" data-ref="n199"><a class="md-toc-inner" href="#header-n199">2 单变量线性回归(Linear Regression with One Variable)</a></span><span class="md-toc-item md-toc-h2" data-ref="n200"><a class="md-toc-inner" href="#header-n200">2.1 模型表示(Model Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n243"><a class="md-toc-inner" href="#header-n243">2.2 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n287"><a class="md-toc-inner" href="#header-n287">2.3 代价函数 - 直观理解1(Cost Function - Intuition I)</a></span><span class="md-toc-item md-toc-h2" data-ref="n309"><a class="md-toc-inner" href="#header-n309">2.4 代价函数 - 直观理解2(Cost Function - Intuition II)</a></span><span class="md-toc-item md-toc-h2" data-ref="n333"><a class="md-toc-inner" href="#header-n333">2.5 梯度下降(Gradient Descent)</a></span><span class="md-toc-item md-toc-h2" data-ref="n363"><a class="md-toc-inner" href="#header-n363">2.6 梯度下降直观理解(Gradient Descent Intuition)</a></span><span class="md-toc-item md-toc-h2" data-ref="n402"><a class="md-toc-inner" href="#header-n402">2.7 线性回归中的梯度下降(Gradient Descent For Linear Regression)</a></span><span class="md-toc-item md-toc-h1" data-ref="n446"><a class="md-toc-inner" href="#header-n446">3 Linear Algebra Review</a></span><span class="md-toc-item md-toc-h2" data-ref="n449"><a class="md-toc-inner" href="#header-n449">3.1 Matrices and Vectors</a></span><span class="md-toc-item md-toc-h2" data-ref="n456"><a class="md-toc-inner" href="#header-n456">3.2 Addition and Scalar Multiplication</a></span><span class="md-toc-item md-toc-h2" data-ref="n463"><a class="md-toc-inner" href="#header-n463">3.3 Matrix Vector Multiplication</a></span><span class="md-toc-item md-toc-h2" data-ref="n470"><a class="md-toc-inner" href="#header-n470">3.4 Matrix Matrix Multiplication</a></span><span class="md-toc-item md-toc-h2" data-ref="n477"><a class="md-toc-inner" href="#header-n477">3.5 Matrix Multiplication Properties</a></span><span class="md-toc-item md-toc-h2" data-ref="n484"><a class="md-toc-inner" href="#header-n484">3.6 Inverse and Transpose</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>1. 引言(Introduction)</h1><h2><a name='header-n3' class='md-header-anchor '></a>1.1 Welcome</h2><p>随着互联网数据不断累积，硬件不断升级迭代，在这个信息爆炸的时代，机器学习已被应用在各行各业中，可谓无处不在。</p><p>一些常见的机器学习的应用，例如：</p><ul><li>手写识别</li><li>垃圾邮件分类</li><li>搜索引擎</li><li>图像处理</li><li>…</li></ul><p>使用到机器学习的一些案例：</p><ul><li><p>数据挖掘</p><ul><li>网页点击流数据分析</li></ul></li><li><p>人工无法处理的工作(量大)</p><ul><li>手写识别</li><li>计算机视觉</li></ul></li><li><p>个人定制</p><ul><li>推荐系统</li></ul></li><li><p>研究大脑</p></li><li><p>……</p></li></ul><h2><a name='header-n57' class='md-header-anchor '></a>1.2 什么是机器学习(What is Machine Learning)</h2><ol start='' ><li><p>机器学习定义
这里主要有两种定义：</p><ul><li><p>Arthur Samuel (1959). Machine Learning: Field of study that gives computers the ability to learn without being explicitly programmed.</p><p>这个定义有点不正式但提出的时间最早，来自于一个懂得计算机编程的下棋菜鸟。他编写了一个程序，但没有显式地编程每一步该怎么走，而是让计算机自己和自己对弈，并不断地计算布局的好坏，来判断什么情况下获胜的概率高，从而积累经验，好似学习，最后，这个计算机程序成为了一个比他自己还厉害的棋手。</p></li><li><p>Tom Mitchell (1998) Well-posed Learning Problem: A computer program is said to learn from experience E with respect to some <strong>task T</strong> and some <strong>performance measure P</strong>, if its performance on T, as measured by P, improves with <strong>experience E</strong>. </p><p>Tom Mitchell 的定义更为现代和正式。在过滤垃圾邮件这个例子中，电子邮件系统会根据用户对电子邮件的标记（是/不是垃圾邮件）不断学习，从而提升过滤垃圾邮件的准确率，定义中的三个字母分别代表：</p><ul><li>T(Task): 过滤垃圾邮件任务。</li><li>P(Performance): 电子邮件系统过滤垃圾邮件的准确率。</li><li>E(Experience): 用户对电子邮件的标记。</li></ul></li></ul></li><li><p>机器学习算法</p><p>主要有两种机器学习的算法分类</p><ol start='' ><li>监督学习</li><li>无监督学习</li></ol><p>两者的区别为<strong>是否需要人工参与数据结果的标注</strong>。这两部分的内容占比很大，并且很重要，掌握好了可以在以后的应用中节省大把大把的时间~</p><p>还有一些算法也属于机器学习领域，诸如：</p><ul><li>半监督学习: 介于监督学习于无监督学习之间</li><li>推荐算法: 没错，就是那些个买完某商品后还推荐同款的某购物网站所用的算法。</li><li>强化学习: 通过观察来学习如何做出动作，每个动作都会对环境有所影响，而环境的反馈又可以引导该学习算法。</li><li>迁移学习</li></ul></li></ol><h2><a name='header-n113' class='md-header-anchor '></a>1.3 监督学习(Supervised Learning)</h2><p>监督学习，即为教计算机如何去完成预测任务（有反馈），预先给一定数据量的输入<strong>和对应的结果</strong>即训练集，建模拟合，最后让计算机预测未知数据的结果。</p><p>监督学习一般有两种：</p><ol start='' ><li><p>回归问题(Regression)</p><p>回归问题即为预测一系列的<strong>连续值</strong>。</p><p>在房屋价格预测的例子中，给出了一系列的房屋面基数据，根据这些数据来预测任意面积的房屋价格。给出照片-年龄数据集，预测给定照片的年龄。</p><p><img src='image/20180105_194712.png' alt='' /></p></li><li><p>分类问题(Classification)</p><p>分类问题即为预测一系列的<strong>离散值</strong>。</p><p>即根据数据预测被预测对象属于哪个分类。</p><p>视频中举了癌症肿瘤这个例子，针对诊断结果，分别分类为良性或恶性。还例如垃圾邮件分类问题，也同样属于监督学习中的分类问题。</p><p><img src='image/20180105_194839.png' alt='' /></p></li></ol><p>视频中提到<strong>支持向量机</strong>这个算法，旨在解决当特征量很大的时候(特征即如癌症例子中的肿块大小，颜色，气味等各种特征)，计算机内存一定会不够用的情况。<strong>支持向量机能让计算机处理无限多个特征。</strong></p><h2><a name='header-n142' class='md-header-anchor '></a>1.4 无监督学习(Unsupervised Learning)</h2><p>相对于监督学习，训练集不会有人为标注的结果（无反馈），我们<strong>不会给出</strong>结果或<strong>无法得知</strong>训练集的结果是什么样，而是单纯由计算机通过无监督学习算法自行分析，从而“得出结果”。计算机可能会把特定的数据集归为几个不同的簇，故叫做聚类算法。</p><p>无监督学习一般分为两种：</p><ol start='' ><li><p>聚类(Clustering)</p><ul><li>新闻聚合</li><li>DNA 个体聚类</li><li>天文数据分析</li><li>市场细分</li><li>社交网络分析</li></ul></li><li><p>非聚类(Non-clustering)</p><ul><li>鸡尾酒问题</li></ul></li></ol><p><strong>新闻聚合</strong></p><p>在例如谷歌新闻这样的网站中，每天后台都会收集成千上万的新闻，然后将这些新闻分组成一个个的新闻专题，这样一个又一个聚类，就是应用了无监督学习的结果。</p><p><strong>鸡尾酒问题</strong></p><p><img src='image/20180105_201639.png' alt='' /></p><p>在鸡尾酒会上，大家说话声音彼此重叠，几乎很难分辨出面前的人说了什么。我们很难对于这个问题进行数据标注，而这里的通过机器学习的无监督学习算法，就可以将说话者的声音同背景音乐分离出来，看视频，效果还不错呢~~。</p><p>嗯，这块是打打鸡血的，只需要一行代码就解决了问题，就是这么简单！当然，我没复现过 ^_^……</p><p>神奇的一行代码：
<code>[W,s,v] = svd((repmat(sum(x.*x,1),size(x,1),1).*x)*x&#39;);</code></p><p><strong>编程语言建议</strong></p><p>在机器学习刚开始时，<strong>推荐使用 Octave 类的工程计算编程软件</strong>，因为在 C++ 或 Java 等编程语言中，编写对应的代码需要用到复杂的库以及要写大量的冗余代码，比较耗费时间，建议可以在学习过后再考虑使用其他语言来构建系统。
另外，在做<strong>原型搭建</strong>的时候也应该先考虑使用类似于 Octave 这种便于计算的编程软件，当其已经可以工作后，才将模型移植到其他的高级编程语言中。</p><blockquote><p>注：Octave 与 MATLAB 语法相近，由于 MATLAB 为商业软件，课程中使用开源且免费的 Octave。</p><p>机器学习领域发展迅速，现在也可使用 Tensorflow 等开源机器学习框架编写机器学习代码，这些框架十分友好，易于编写及应用。</p></blockquote><h1><a name='header-n199' class='md-header-anchor '></a>2 单变量线性回归(Linear Regression with One Variable)</h1><h2><a name='header-n200' class='md-header-anchor '></a>2.1 模型表示(Model Representation)</h2><ol start='' ><li><p>房价预测训练集</p><figure><table><thead><tr><th>Size in <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.339ex" height="2.757ex" viewBox="0 -889.4 2298.9 1187.2" role="img" focusable="false" style="vertical-align: -0.692ex;"><defs><path stroke-width="1" id="E1-MJMATHI-66" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path stroke-width="1" id="E1-MJMATHI-65" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 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xlink:href="#E12-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-12">x</script>)</th><th>Price ($) in 1000&#39;s(<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-27-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.155ex" height="1.877ex" viewBox="0 -504.6 497.5 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E27-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E27-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-27">y</script>)</th></tr></thead><tbody><tr><td>2104</td><td>460</td></tr><tr><td>1416</td><td>232</td></tr><tr><td>1534</td><td>315</td></tr><tr><td>852</td><td>178</td></tr><tr><td>...</td><td>...</td></tr></tbody></table></figure></li></ol><p>房价预测训练集中，同时给出了输入 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-12-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.41ex" viewBox="0 -504.6 572.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E12-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use 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-167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E27-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-27">y</script>，即给出了人为标注的<strong>”正确结果“</strong>，且预测的量是连续的，属于监督学习中的回归问题。</p><ol start='2' ><li><strong>问题解决模型</strong></li></ol><p><img src='image/20180105_212048.png' alt='' /></p><p>其中 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-43-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.339ex" height="1.994ex" viewBox="0 -755.9 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stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-3D" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-30" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-2B" x="4839" y="0"></use><g transform="translate(5840,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-31" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-78" x="6763" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-73">h_\theta(x)=\theta_0+\theta_1x</script>，为解决房价问题的一种可行表达式。</p><blockquote><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-12-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.41ex" viewBox="0 -504.6 572.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E12-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E12-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-12">x</script>: 特征/输入变量。</p></blockquote><p>上式中，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-70-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-70">\theta</script> 为参数，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-70-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-70">\theta</script> 的变化才决定了输出结果，不同以往，这里的 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-12-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.33ex" height="1.41ex" viewBox="0 -504.6 572.5 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E12-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 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254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-70">\theta</script> 以更好地拟合数据，成了求解该问题的最终问题。</p><p>单变量，即只有一个特征(如例子中房屋的面积这个特征)。</p><h2><a name='header-n243' class='md-header-anchor '></a>2.2 代价函数(Cost Function)</h2><blockquote><p>李航《统计学习方法》一书中，损失函数与代价函数两者为<strong>同一概念</strong>，未作细分区别，全书没有和《深度学习》一书一样混用，而是统一使用<strong>损失函数</strong>来指代这类类似概念。</p><p>吴恩达(Andrew Ng)老师在其公开课中对两者做了细分。<strong>如果要听他的课做作业，不细分这两个概念是会被打小手扣分的</strong>！这也可能是因为老师发现了业内混用的乱象，想要治一治吧。</p><p><strong>损失函数</strong>(Loss/Error Function): 计算<strong>单个</strong>训练集的误差</p><p><strong>代价函数</strong>(Cost Function): 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inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-70">\theta</script> 的取值，则问题可表达为<strong>求解 <span class="MathJax_Preview"></span><span 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568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E29-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="473" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E29-MJMAIN-31" x="84" y="572"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E29-MJMAIN-32" x="84" y="-532"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-29">\frac{1}{2}</script> 。</p></blockquote><p>讨论到这里，我们的问题就转化成了<strong>求解 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-30-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.991ex" height="2.757ex" viewBox="0 -840.1 3871.1 1187.2" role="img" focusable="false" style="vertical-align: -0.806ex;"><defs><path stroke-width="1" id="E30-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E30-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E30-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E30-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E30-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="1" id="E30-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E30-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E30-MJMAIN-31" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E30-MJMAIN-29" x="2681" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-30">J\left( \theta_0, \theta_1  \right)</script> 的最小值</strong>。</p><h2><a name='header-n287' class='md-header-anchor '></a>2.3 代价函数 - 直观理解1(Cost Function - Intuition I)</h2><p>根据上节视频，列出如下定义：</p><ul><li>假设函数(Hypothesis): <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-73-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="17.039ex" height="2.577ex" viewBox="0 -806.1 7336.3 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E73-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E73-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E73-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E73-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E73-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 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id="E73-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-3B8" x="815" y="-219"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-28" x="1008" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-78" x="1397" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-29" x="1970" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-3D" x="2637" y="0"></use><g transform="translate(3694,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-30" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-2B" x="4839" y="0"></use><g transform="translate(5840,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMAIN-31" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E73-MJMATHI-78" x="6763" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-73">h_\theta(x)=\theta_0+\theta_1x</script></li><li>参数(Parameters): <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-47-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.323ex" height="2.461ex" viewBox="0 -806.1 2292 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E47-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E47-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E47-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="1" id="E47-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E47-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E47-MJMAIN-30" x="663" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E47-MJMAIN-2C" x="923" y="0"></use><g transform="translate(1368,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E47-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E47-MJMAIN-31" x="663" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-47">\theta_0, \theta_1</script></li><li>代价函数(Cost Function): <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-74-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="36.164ex" height="5.846ex" viewBox="0 -1459.5 15570.8 2517" role="img" focusable="false" style="vertical-align: -2.456ex;"><defs><path stroke-width="1" id="E74-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E74-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E74-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E74-MJMAIN-30" d="M96 585Q152 666 249 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xlink:href="#E34-MJMAIN-30" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E34-MJMAIN-2C" x="1312" y="0"></use><g transform="translate(1758,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E34-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E34-MJMAIN-31" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E34-MJMAIN-29" x="2681" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-34">\underset{\theta_0, \theta_1}{\text{minimize}} J \left(\theta_0, \theta_1 \right)</script></li></ul><p>为了直观理解代价函数到底是在做什么，先假设 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-35-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.406ex" height="2.344ex" viewBox="0 -806.1 2758 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E35-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E35-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E35-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E35-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E35-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E35-MJMAIN-31" x="663" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E35-MJMAIN-3D" x="1201" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E35-MJMAIN-30" x="2257" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-35">\theta_1 = 0</script>，并假设训练集有三个数据，分别为<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-36-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="18.347ex" height="2.577ex" viewBox="0 -806.1 7899.2 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E36-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E36-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" 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549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E36-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-31" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-2C" x="890" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-31" x="1335" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-29" x="1835" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-2C" x="2391" y="0"></use><g transform="translate(2837,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-32" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-2C" x="890" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-32" x="1335" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-29" x="1835" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-2C" x="5228" y="0"></use><g transform="translate(5674,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-33" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-2C" x="890" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-33" x="1335" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E36-MJMAIN-29" x="1835" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-36">\left(1, 1\right), \left(2, 2\right), \left(3, 3\right)</script>，这样在平面坐标系中绘制出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-51-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E51-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="1" id="E51-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E51-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E51-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="1" id="E51-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E51-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E51-MJMATHI-3B8" x="815" y="-219"></use><g transform="translate(1175,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E51-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E51-MJMATHI-78" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E51-MJMAIN-29" x="962" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-51">h_\theta\left(x\right)</script> ，并分析 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-39-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.991ex" height="2.577ex" viewBox="0 -806.1 3871.1 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E39-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E39-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E39-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E39-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 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y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-39">J\left(\theta_0, \theta_1\right)</script> 的变化。</p><p><img src='image/20180106_085915.png' alt='' /></p><p>右图 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-39-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.991ex" height="2.577ex" viewBox="0 -806.1 3871.1 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E39-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path 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525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E39-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="1" id="E39-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E39-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 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xlink:href="#E39-MJMAIN-29" x="2681" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-39">J\left(\theta_0, \theta_1\right)</script> 随着 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-69-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.344ex" viewBox="0 -806.1 923.4 1009.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="1" id="E69-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E69-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E69-MJMAIN-31" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-69">\theta_1</script> 的变化而变化，可见<strong>当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-41-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.406ex" height="2.414ex" viewBox="0 -790.8 2758 1039.3" role="img" focusable="false" style="vertical-align: -0.577ex;"><defs><path stroke-width="1" id="E41-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E41-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E41-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></defs><g 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-30" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-2C" x="1312" y="0"></use><g transform="translate(1758,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-31" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-29" x="2681" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-3D" x="4148" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E42-MJMAIN-30" x="5205" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-42">J\left(\theta_0, \theta_1 \right) = 0</script>，取得最小值，</strong>对应于左图青色直线，即函数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-43-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.339ex" height="1.994ex" viewBox="0 -755.9 576.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E43-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E43-MJMATHI-68" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-43">h</script> 拟合程度最好的情况。</p><h2><a name='header-n309' class='md-header-anchor '></a>2.4 代价函数 - 直观理解2(Cost Function - Intuition II)</h2><blockquote><p>注：该部分由于涉及到了多变量成像，可能较难理解，要求只需要理解上节内容即可，该节如果不能较好理解可跳过。</p></blockquote><p>给定数据集：</p><p><img src='image/20180106_091307.png' alt='' /></p><p>参数在 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-44-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.145ex" height="2.461ex" viewBox="0 -806.1 923.4 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E44-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E44-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E44-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E44-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex" id="MathJax-Element-44">\theta_0</script> 不恒为 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-45-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.162ex" height="1.994ex" viewBox="0 -755.9 500.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E45-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E45-MJMAIN-30" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-45">0</script> 时代价函数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-67-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E67-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E67-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E67-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E67-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E67-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E67-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E67-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E67-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-67">J\left(\theta\right)</script> 关于 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-47-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.323ex" height="2.461ex" viewBox="0 -806.1 2292 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E47-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E47-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E47-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="1" id="E47-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E47-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E47-MJMAIN-30" x="663" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E47-MJMAIN-2C" x="923" y="0"></use><g transform="translate(1368,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E47-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E47-MJMAIN-31" x="663" y="-213"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-47">\theta_0, \theta_1</script> 的3-D图像，图像中的高度为代价函数的值。</p><p><img src='image/20180106_090904.png' alt='' /></p><p>由于3-D图形不便于标注，所以将3-D图形转换为<strong>轮廓图(contour plot)</strong>，下面用轮廓图（下图中的右图）来作直观理解，其中相同颜色的一个圈代表着同一高度（同一 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-67-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E67-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E67-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="1" id="E67-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E67-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E67-MJMATHI-4A" x="0" y="0"></use><g transform="translate(800,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E67-MJMAIN-28" x="0" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E67-MJMATHI-3B8" x="389" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E67-MJMAIN-29" x="859" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-67">J\left(\theta\right)</script> 值）。</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-49-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.17ex" height="2.461ex" viewBox="0 -806.1 6962.1 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E49-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E49-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E49-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E49-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="1" id="E49-MJMAIN-36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path><path stroke-width="1" id="E49-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="1" id="E49-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMAIN-30" x="663" y="-213"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMAIN-3D" x="1201" y="0"></use><g transform="translate(2257,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMAIN-33"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMAIN-36" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMAIN-30" x="1001" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMAIN-2C" x="3758" y="0"></use><g transform="translate(4204,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMAIN-31" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMAIN-3D" x="5405" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E49-MJMAIN-30" x="6461" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-49">\theta_0 = 360, \theta_1 =0</script> 时：</p><p><img src='image/0f38a99c8ceb8aa5b90a5f12136fdf43.png' alt='' /></p><p>大概在 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-50-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="19.142ex" height="2.461ex" viewBox="0 -806.1 8241.6 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="1" id="E50-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="1" id="E50-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="1" id="E50-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E50-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="1" id="E50-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="1" id="E50-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="1" id="E50-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 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xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-3D" x="1201" y="0"></use><g transform="translate(2257,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-30"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-2E" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-31" x="779" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-32" x="1279" y="0"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-2C" x="4037" y="0"></use><g transform="translate(4482,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-31" x="663" y="-213"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-3D" x="5683" y="0"></use><g transform="translate(6740,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-32"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-35" x="500" y="0"></use><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E50-MJMAIN-30" x="1001" y="0"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-50">\theta_0 = 0.12, \theta_1 =250</script> 时：</p><p><img src='image/20180106_092119.png' alt='' /></p><p>上图中最中心的点（红点），近乎为图像中的最低点，也即代价函数的最小值，此时对应 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-51-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.868ex" height="2.577ex" viewBox="0 -806.1 2526.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E51-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 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'></a>2.5 梯度下降(Gradient Descent)</h2><p>在特征量很大的情况下，即便是借用计算机来生成图像，人工的方法也很难读出 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-67-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.758ex" height="2.577ex" viewBox="0 -806.1 2048.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="1" id="E67-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="1" id="E67-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 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minimum)</strong>，由于下降的情况只考虑当前参数组合周围的情况，所以无法确定当前的局部最小值是否就是<strong>全局最小值(global minimum)</strong>，不同的初始参数组合，可能会产生不同的局部最小值。</p><p>下图根据不同的起始点，产生了两个不同的局部最小值。</p><p><img src='image/db48c81304317847870d486ba5bb2015.jpg' alt='' /></p><p>视频中举了下山的例子，即我们在山顶上的某个位置，为了下山，就不断地看一下周围<strong>下一步往哪走</strong>下山比较快，然后就<strong>迈出那一步</strong>，一直重复，直到我们到达山下的某一处<strong>陆地</strong>。</p><p>梯度下降公式：</p><p><span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-75-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="28.85ex" height="12.149ex" viewBox="0 -2866.8 12421.4 5231" role="img" focusable="false" style="vertical-align: -5.491ex;"><defs><path stroke-width="1" id="E75-MJMAIN-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 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type="math/tex" id="MathJax-Element-61">J\left( \theta_0, \theta_1 \right)</script> 的偏导</p></blockquote><p>公式中，学习速率决定了参数值变化的速率即”<strong>走多少距离</strong>“，而偏导这部分决定了下降的方向即”<strong>下一步往哪里</strong>“走（当然实际上的走多少距离是由偏导值给出的，学习速率起到调整后决定的作用），收敛处的局部最小值又叫做极小值，即”<strong>陆地</strong>“。</p><p><img src='image/20180106_101659.png' alt='' /></p><p>注意，在计算时要<strong>批量更新 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-70-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.09ex" height="2.11ex" viewBox="0 -806.1 469.5 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 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transform="translate(5134,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMAIN-29" x="734" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMAIN-31" x="809" y="-435"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMAIN-2212" x="6824" y="0"></use><g transform="translate(7825,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E79-MJSZ2-29" x="9220" y="-1"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-79">\left(\frac{1}{m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} \right)*\frac{\partial}{\partial\theta_j}{{\left(\theta_0{x_0^{(i)}} + \theta_1{x_1^{(i)}}-{{y}^{(i)}} \right)}}</script></p><p>所以当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-80-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.246ex" height="2.461ex" viewBox="-11.5 -755.9 2258.6 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.027ex;"><defs><path stroke-width="1" id="E80-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path><path stroke-width="1" id="E80-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="1" id="E80-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 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y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMATHI-3B8" x="815" y="-219"></use></g><g transform="translate(1633,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJSZ1-28"></use><g transform="translate(458,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJSZ1-29" x="1926" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMAIN-2212" x="4240" y="0"></use><g transform="translate(5241,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,362)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMAIN-29" x="734" y="0"></use></g></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJSZ1-29" x="6636" y="-1"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMAIN-2217" x="14410" y="0"></use><g transform="translate(15132,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMAIN-29" x="734" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E81-MJMAIN-30" x="809" y="-435"></use></g></g></svg></span><script type="math/tex" id="MathJax-Element-81">\frac{\partial}{\partial\theta_0} J(\theta)=\frac{1}{m}\sum\limits_{i=1}^{m}{{\left( {{h}_{\theta }}\left( {{x}^{(i)}} \right)-{{y}^{(i)}} \right)}} *x_0^{(i)}</script></p><p>所以当 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-82-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.246ex" height="2.461ex" viewBox="-11.5 -755.9 2258.6 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: 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400 -430T690 -924Q701 -936 701 -940Z"></path><path stroke-width="1" id="E84-MJSZ3-29" d="M34 1438Q34 1446 37 1448T50 1450H56H71Q73 1448 99 1423T144 1380T198 1319T260 1238T323 1137T385 1013T440 864T485 688T514 485T526 251Q526 134 519 53Q472 -519 162 -860Q139 -885 119 -904T86 -936T71 -949H56Q43 -949 39 -947T34 -937Q88 -883 140 -813Q428 -430 428 251Q428 453 402 628T338 922T245 1146T145 1309T46 1425Q44 1427 42 1429T39 1433T36 1436L34 1438Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJSZ3-28"></use><g transform="translate(736,0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJSZ1-2211" x="46" y="0"></use><g transform="translate(0,-890)"><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMAIN-3D" x="345" y="0"></use><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMAIN-31" x="1124" y="0"></use></g><use transform="scale(0.707)" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJMATHI-6D" x="372" y="1344"></use></g><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E84-MJSZ3-29" x="1885" y="-1"></use></g></svg></span><script type="math/tex" id="MathJax-Element-84">\left(\sum\limits_{i=1}^{m}\right)</script> 更新。</p><p>由于线性回归函数呈现<strong>碗状</strong>，且<strong>只有一个</strong>全局的最优值，所以函数<strong>一定总会</strong>收敛到全局最小值（学习速率不可过大）。同时，函数 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-85-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.471ex" height="1.994ex" viewBox="0 -755.9 633.5 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="1" id="E85-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#E85-MJMATHI-4A" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-85">J</script> 被称为<strong>凸二次函数</strong>，而线性回归函数求解最小值问题属于<strong>凸函数优化问题</strong>。</p><p><img src='image/24e9420f16fdd758ccb7097788f879e7.png' alt='' /></p><p>另外，使用循环求解，代码较为冗余，后面会讲到如何使用<strong>向量化(Vectorization)</strong>来简化代码并优化计算，使梯度下降运行的更快更好。</p><h1><a name='header-n446' class='md-header-anchor '></a>3 Linear Algebra Review</h1><p>这部分，学过线性代数的可以复习一下，比较基础。笔记整理暂留。</p><h2><a name='header-n449' class='md-header-anchor '></a>3.1 Matrices and Vectors</h2><p>Octave/Matlab 代码:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="matlab" style="break-inside: unset;"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>17</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% The ; denotes we are going back to a new row.</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>, <span class="cm-number">2</span>, <span class="cm-number">3</span>; <span class="cm-number">4</span>, <span class="cm-number">5</span>, <span class="cm-number">6</span>; <span class="cm-number">7</span>, <span class="cm-number">8</span>, <span class="cm-number">9</span>; <span class="cm-number">10</span>, <span class="cm-number">11</span>, <span class="cm-number">12</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize a vector </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">v</span> = [<span class="cm-number">1</span>;<span class="cm-number">2</span>;<span class="cm-number">3</span>] </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Get the dimension of the matrix A where m = rows and n = columns</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">[<span class="cm-variable">m</span>,<span class="cm-variable">n</span>] = <span class="cm-builtin">size</span>(<span class="cm-variable">A</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% You could also store it this way</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">dim_A</span> = <span class="cm-builtin">size</span>(<span class="cm-variable">A</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Get the dimension of the vector v </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">dim_v</span> = <span class="cm-builtin">size</span>(<span class="cm-variable">v</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Now let's index into the 2nd row 3rd column of matrix A</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A_23</span> = <span class="cm-variable">A</span>(<span class="cm-number">2</span>,<span class="cm-number">3</span>)</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 381px;"></div><div class="CodeMirror-gutters" style="height: 411px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><p>执行结果:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="" style="break-inside: unset;"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>24</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  1 &nbsp;  2 &nbsp;  3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  4 &nbsp;  5 &nbsp;  6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  7 &nbsp;  8 &nbsp;  9</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 10 &nbsp; 11 &nbsp; 12</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">v =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">m =  4</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">n =  3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">dim_A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 4 &nbsp; 3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">dim_v =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 3 &nbsp; 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A_23 =  6</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 538px;"></div><div class="CodeMirror-gutters" style="height: 568px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><h2><a name='header-n456' class='md-header-anchor '></a>3.2 Addition and Scalar Multiplication</h2><p>Octave/Matlab 代码:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="matlab" style="break-inside: unset;"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>21</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize matrix A and B </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>, <span class="cm-number">2</span>, <span class="cm-number">4</span>; <span class="cm-number">5</span>, <span class="cm-number">3</span>, <span class="cm-number">2</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">B</span> = [<span class="cm-number">1</span>, <span class="cm-number">3</span>, <span class="cm-number">4</span>; <span class="cm-number">1</span>, <span class="cm-number">1</span>, <span class="cm-number">1</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize constant s </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">s</span> = <span class="cm-number">2</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% See how element-wise addition works</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">add_AB</span> = <span class="cm-variable">A</span> <span class="cm-operator">+</span> <span class="cm-variable">B</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% See how element-wise subtraction works</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">sub_AB</span> = <span class="cm-variable">A</span> <span class="cm-operator">-</span> <span class="cm-variable">B</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% See how scalar multiplication works</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">mult_As</span> = <span class="cm-variable">A</span> <span class="cm-operator">*</span> <span class="cm-variable">s</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Divide A by s</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">div_As</span> = <span class="cm-variable">A</span> <span class="cm-operator">/</span> <span class="cm-variable">s</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% What happens if we have a Matrix + scalar?</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">add_As</span> = <span class="cm-variable">A</span> <span class="cm-operator">+</span> <span class="cm-variable">s</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 470px;"></div><div class="CodeMirror-gutters" style="height: 500px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><p>执行结果:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="" style="break-inside: unset;"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>35</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 2 &nbsp; 4</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 5 &nbsp; 3 &nbsp; 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">B =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 3 &nbsp; 4</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 1 &nbsp; 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">s =  2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">add_AB =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 2 &nbsp; 5 &nbsp; 8</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 6 &nbsp; 4 &nbsp; 3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">sub_AB =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 0  -1 &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 4 &nbsp; 2 &nbsp; 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">mult_As =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  2 &nbsp;  4 &nbsp;  8</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 10 &nbsp;  6 &nbsp;  4</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">div_As =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 0.50000 &nbsp; 1.00000 &nbsp; 2.00000</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 2.50000 &nbsp; 1.50000 &nbsp; 1.00000</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">add_As =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 3 &nbsp; 4 &nbsp; 6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 7 &nbsp; 5 &nbsp; 4</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 784px;"></div><div class="CodeMirror-gutters" style="height: 814px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><h2><a name='header-n463' class='md-header-anchor '></a>3.3 Matrix Vector Multiplication</h2><p>Octave/Matlab 代码:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="matlab"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize matrix A </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>, <span class="cm-number">2</span>, <span class="cm-number">3</span>; <span class="cm-number">4</span>, <span class="cm-number">5</span>, <span class="cm-number">6</span>;<span class="cm-number">7</span>, <span class="cm-number">8</span>, <span class="cm-number">9</span>] </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize vector v </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">v</span> = [<span class="cm-number">1</span>; <span class="cm-number">1</span>; <span class="cm-number">1</span>] </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Multiply A * v</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">Av</span> = <span class="cm-variable">A</span> <span class="cm-operator">*</span> <span class="cm-variable">v</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 224px;"></div><div class="CodeMirror-gutters" style="height: 254px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><p>执行结果:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="" style="break-inside: unset;"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>18</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 2 &nbsp; 3</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 4 &nbsp; 5 &nbsp; 6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 7 &nbsp; 8 &nbsp; 9</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">v =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">Av =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 15</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 24</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 403px;"></div><div class="CodeMirror-gutters" style="height: 433px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><h2><a name='header-n470' class='md-header-anchor '></a>3.4 Matrix Matrix Multiplication</h2><p>Octave/Matlab 代码:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="matlab"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>10</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize a 3 by 2 matrix </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>, <span class="cm-number">2</span>; <span class="cm-number">3</span>, <span class="cm-number">4</span>;<span class="cm-number">5</span>, <span class="cm-number">6</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize a 2 by 1 matrix </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">B</span> = [<span class="cm-number">1</span>; <span class="cm-number">2</span>] </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% We expect a resulting matrix of (3 by 2)*(2 by 1) = (3 by 1) </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">mult_AB</span> = <span class="cm-variable">A</span><span class="cm-operator">*</span><span class="cm-variable">B</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Make sure you understand why we got that result</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 224px;"></div><div class="CodeMirror-gutters" style="height: 254px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><p>执行结果:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="" style="break-inside: unset;"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>17</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 3 &nbsp; 4</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 5 &nbsp; 6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">B =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">mult_AB =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 11</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 17</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 381px;"></div><div class="CodeMirror-gutters" style="height: 411px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><h2><a name='header-n477' class='md-header-anchor '></a>3.5 Matrix Multiplication Properties</h2><p>Octave/Matlab 代码:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="matlab" style="break-inside: unset;"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>22</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize random matrices A and B </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>,<span class="cm-number">2</span>;<span class="cm-number">4</span>,<span class="cm-number">5</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">B</span> = [<span class="cm-number">1</span>,<span class="cm-number">1</span>;<span class="cm-number">0</span>,<span class="cm-number">2</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize a 2 by 2 identity matrix</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">I</span> = <span class="cm-builtin">eye</span>(<span class="cm-number">2</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% The above notation is the same as I = [1,0;0,1]</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% What happens when we multiply I*A ? </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">IA</span> = <span class="cm-variable">I</span><span class="cm-operator">*</span><span class="cm-variable">A</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% How about A*I ? </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">AI</span> = <span class="cm-variable">A</span><span class="cm-operator">*</span><span class="cm-variable">I</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Compute A*B </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">AB</span> = <span class="cm-variable">A</span><span class="cm-operator">*</span><span class="cm-variable">B</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Is it equal to B*A? </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">BA</span> = <span class="cm-variable">B</span><span class="cm-operator">*</span><span class="cm-variable">A</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Note that IA = AI but AB != BA</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 493px;"></div><div class="CodeMirror-gutters" style="height: 523px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><p>执行结果:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="" style="break-inside: unset;"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>36</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 4 &nbsp; 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">B =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 0 &nbsp; 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">I =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">Diagonal Matrix</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 0 &nbsp; 1</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">IA =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 4 &nbsp; 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">AI =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 2</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 4 &nbsp; 5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">AB =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  1 &nbsp;  5</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  4 &nbsp; 14</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">BA =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  5 &nbsp;  7</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">36</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp;  8 &nbsp; 10</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 806px;"></div><div class="CodeMirror-gutters" style="height: 836px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><h2><a name='header-n484' class='md-header-anchor '></a>3.6 Inverse and Transpose</h2><p>Octave/Matlab 代码:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="matlab"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>13</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Initialize matrix A </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A</span> = [<span class="cm-number">1</span>,<span class="cm-number">2</span>,<span class="cm-number">0</span>;<span class="cm-number">0</span>,<span class="cm-number">5</span>,<span class="cm-number">6</span>;<span class="cm-number">7</span>,<span class="cm-number">0</span>,<span class="cm-number">9</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Transpose A </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A_trans</span> = <span class="cm-variable">A</span><span class="cm-operator">'</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% Take the inverse of A </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A_inv</span> = <span class="cm-builtin">inv</span>(<span class="cm-variable">A</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">% What is A^(-1)*A? </span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">A_invA</span> = <span class="cm-builtin">inv</span>(<span class="cm-variable">A</span>)<span class="cm-operator">*</span><span class="cm-variable">A</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 291px;"></div><div class="CodeMirror-gutters" style="height: 321px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre><p>执行结果:</p><pre class="md-fences md-end-block md-fences-with-lineno" lang="" style="break-inside: unset;"> <div class="CodeMirror cm-s-inner CodeMirror-wrap"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 39px;"></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 35px; margin-bottom: 0px; border-right-width: 30px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>24</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -35px; width: 35px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 2 &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 0 &nbsp; 5 &nbsp; 6</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 7 &nbsp; 0 &nbsp; 9</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A_trans =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1 &nbsp; 0 &nbsp; 7</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 2 &nbsp; 5 &nbsp; 0</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 0 &nbsp; 6 &nbsp; 9</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A_inv =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 0.348837  -0.139535 &nbsp; 0.093023</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 0.325581 &nbsp; 0.069767  -0.046512</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">  -0.271318 &nbsp; 0.108527 &nbsp; 0.038760</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">A_invA =</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 1.00000  -0.00000 &nbsp; 0.00000</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> &nbsp; 0.00000 &nbsp; 1.00000  -0.00000</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 26px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">  -0.00000 &nbsp; 0.00000 &nbsp; 1.00000</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -35px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 26px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="">​</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 30px; width: 1px; border-bottom: 0px solid transparent; top: 538px;"></div><div class="CodeMirror-gutters" style="height: 568px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 34px;"></div></div></div></div></pre></div>
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